2025/05/09 by Andrea Pedrini, Pedrini, Andrea, Epifanio G. Virga +1 · 1 citation
Engineering · Mathematics · #Advanced Materials and Mechanics #Algebraic and Geometric Analysis #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Soft Condensed Matter (cond-mat.soft)
paper · pdf · doi:10.48550/arxiv.2505.06037
openalex publication_date 2025/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An ant-like observer confined to a two-dimensional surface traversed by stripes would wonder whether this striped landscape could be devised in such a way as to appear to be the same wherever they go. Differently stated, this is the problem studied in this paper. In a more technical jargon, we determine all possible uniform nematic fields on a smooth surface. It was already known that for such a field to exist, the surface must have constant negative Gaussian curvature. Here, we show that all uniform nematic fields on such a surface are parallel transported (in Levi-Civita's sense) by special systems of geodesics, which (with scant inventiveness) are termed uniform. We prove that, for every geodesic on the surface, there are two systems of uniform geodesics that include it; they are conventionally called right and left, to allude at a possible intrinsic definition of handedness. We found explicitly all uniform fields for Beltrami's pseudosphere. Since both geodesics and uniformity are preserved under isometries, by a classical theorem of Minding, the solution for the pseudosphere carries over all other admissible surfaces, thus providing a general solution to the problem (at least in principle).